
any \(H^\infty \) -supermanifold is a \(G^\infty \) - or a \(GH^\infty \) -supermanifold, a consequence of the trivial facts that any \(H^\infty \) -function is a \(GH^\infty \) -function, and any \(GH^\infty \) -function is a \(G^\infty \) -function;

conversely any \(G^\infty \) -supermanifold can be endowed with a (non unique) \(H^\infty \) -superstructure, a consequence of 3. and 4. below;

any \(G^\infty \) -supermanifold has a split structure (i.e. is of the form \(S(|\mathcal {M}|,E)\) , where \(E\)  is a vector bundle over \(|\mathcal {M}|\) ) [1]};

any split manifold \(S(|\mathcal {M}|,E)\)  has a \(H^\infty \) -structure;

any \(H^\infty \) -supermanifold is a (skeletal) algebro-geometric supermanifold [2]};

conversely any (skeletal) algebro-geometric supermanifold is a \(H^\infty \) -supermanifold [3]}.

